Subgroups — C₂₃×#12886 #19160512

Order 5888.

302 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1383	C₂	1	4	yes	maximal
5	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
6	C₂×C₂₃×#108	—	2	1	no	—
7	C₂×C₂₃×#108	C₂²	1	2	yes	—
8	C₂×C₂₃×#106	—	2	1	no	—
9	C₂×C₂₃×#106	C₂²	1	2	yes	—
10	C₂₃×#423	—	2	1	no	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#384	—	2	1	no	—
13	C₂₃×#346	C₄	1	2	yes	complemented
14	C₄×C₁₆×C₂₃	C₄	1	2	yes	complemented
15	C₂×M₁₆×C₂₃	—	2	2	no	—
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
19	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
20	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
21	C₂₃×M₃₂	—	2	2	no	—
22	C₂₃×M₃₂	—	2	2	no	—
23	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
24	C₂₃×#108	—	2	4	no	—
25	C₂₃×#108	—	4	2	no	—
26	C₂₃×#106	—	2	4	no	—
27	C₂₃×#106	—	4	2	no	—
28	C₂₃×C₄⋊C₈	—	4	2	no	—
29	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	—	4	2	no	—
34	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	M₁₆×C₂₃	—	4	2	no	—
37	M₁₆×C₂₃	—	4	2	no	—
38	C₂×C₈×C₂₃	—	4	2	no	—
39	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	D₈	1	2	yes	—
42	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
43	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₂₃	#41	1	2	yes	—
46	C₂×C₈×C₂₃	#41	1	2	yes	—
47	C₄⋊C₄×C₂₃	—	4	2	no	—
48	C₄⋊C₄×C₂₃	—	4	2	no	—
49	C₄⋊C₄×C₂₃	—	8	4	no	—
50	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₂₃	SD₁₆	1	2	yes	—
53	C₁₆×C₂₃	—	2	4	no	—
54	#12886	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	—	8	4	no	—
58	C₂×C₄×C₂₃	—	8	2	no	—
59	C₂×C₄×C₂₃	#106	1	2	yes	—
60	C₂×C₄×C₂₃	#106	1	2	yes	—
61	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₈×C₂₃	—	8	2	no	—
68	C₈×C₂₃	#107	1	2	yes	—
69	C₈×C₂₃	#107	1	2	yes	—
70	C₈×C₂₃	#117	1	2	yes	—
71	C₈×C₂₃	#125	1	2	yes	—
72	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
73	#1848	C₂×C₂₃	1	1	yes	characteristic
74	#1844	C₂×C₂₃	1	1	yes	characteristic
75	#1383	C₂×C₂₃	1	4	yes	—
76	C₂²×C₂₃	—	2	2	no	—
77	C₂²×C₂₃	C₂×#106	1	1	yes	characteristic
78	C₂²×C₂₃	#486	1	1	yes	characteristic
79	C₂²×C₂₃	#487	1	1	yes	characteristic
80	C₄×C₂₃	—	2	2	no	—
81	C₄×C₂₃	—	16	2	no	—
82	C₄×C₂₃	#365	1	4	yes	—
83	C₄×C₂₃	#382	1	1	yes	characteristic
84	C₄×C₂₃	C₂×#107	1	1	yes	characteristic
85	C₄×C₂₃	#486	1	1	yes	characteristic
86	C₄×C₂₃	#487	1	1	yes	characteristic
87	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
88	C₂×#108	—	2	1	no	—
89	C₂×#108	C₂²×C₂₃	1	2	yes	—
90	C₂×#106	—	2	1	no	—
91	C₂×#106	C₂²×C₂₃	1	2	yes	—
92	#423	—	2	1	no	—
93	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
94	#384	—	2	1	no	—
95	#346	C₄×C₂₃	1	2	yes	complemented
96	C₄×C₁₆	C₄×C₂₃	1	2	yes	complemented
97	C₂×C₂₃	—	2	1	no	—
98	C₂×C₂₃	—	2	1	no	—
99	C₂×C₂₃	#1834	1	1	yes	characteristic
100	C₂×C₂₃	C₂×#365	1	2	yes	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
104	C₂×C₄⋊C₄	—	2	2	no	—
105	C₂×C₄⋊C₄	—	4	2	no	—
106	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	2	2	no	—
109	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
110	#108	—	2	4	no	—
111	#108	—	4	2	no	—
112	#106	—	2	4	no	—
113	#106	—	4	2	no	—
114	C₄⋊C₈	—	4	2	no	—
115	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
116	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
117	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
118	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
119	C₂₃	#12886	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
122	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
127	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
128	C₂×C₈	D₈×C₂₃	1	2	yes	—
129	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
130	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
131	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
132	C₂×C₈	#41×C₂₃	1	2	yes	—
133	C₂×C₈	#41×C₂₃	1	2	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	8	4	no	—
137	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
138	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
139	C₄²	SD₁₆×C₂₃	1	2	yes	—
140	C₁₆	—	2	4	no	—
141	C₂³	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	C₂₃×#106	1	2	yes	—
146	C₂×C₄	C₂₃×#106	1	2	yes	—
147	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
148	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
149	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
150	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
152	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
153	C₈	—	8	2	no	—
154	C₈	C₂₃×#107	1	2	yes	—
155	C₈	C₂₃×#107	1	2	yes	—
156	C₈	C₂₃×#117	1	2	yes	—
157	C₈	C₂₃×#125	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₂×C₂₃×#106	1	1	yes	characteristic
160	C₂²	C₂₃×#486	1	1	yes	characteristic
161	C₂²	C₂₃×#487	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	16	2	no	—
164	C₄	C₂₃×#365	1	4	yes	—
165	C₄	C₂₃×#382	1	1	yes	characteristic
166	C₄	C₂×C₂₃×#107	1	1	yes	characteristic
167	C₄	C₂₃×#486	1	1	yes	characteristic
168	C₄	C₂₃×#487	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	C₂₃×#1834	1	1	yes	characteristic
172	C₂	C₂×C₂₃×#365	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂³×C₂₃	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
13. C₂₃×#346	C₄	semidirect
14. C₄×C₁₆×C₂₃	C₄	semidirect
54. #12886	C₂₃	direct (N × H)
95. #346	C₄×C₂₃	semidirect
96. C₄×C₁₆	C₄×C₂₃	semidirect
119. C₂₃	#12886	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₂₃×M₃₂	2
C₂₃×#1848	2
C₂₃×#1844	2
C₂₃×#1383	2
#12886	23

Subgroups — C23×#12886 #19160512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12886 #19160512